Question

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2+y^2=16$ by the lines, $x+y=n;n\in N$, where $N$ is the set of all the natural numbers, is :

• A. $105$
• B. $210$
• C. $320$
• D. $160$

Answer 1

The correct option is B $210$

Equation of the given circle,
$x^2+y^2=16$
Equation of chord
$x+y=n;$ where $n\in N$
Length of chord$=2\sqrt{16-\frac{n^2}{2}}$
Squaring both the sides,
${\text{(Length of chord)}}^2=2(32-n^2)$
Possible values of $n=1,2,3,4,5$

Sum of the squares of the lengths of the chords
$=2\left(31\right)+2\left(28\right)+2\left(23\right)+2\left(16\right)+2\left(7\right)$
$=210$